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An effective high order interpolation scheme in BIEMs
for biharmonic boundary value problems
N. Mai-Duy∗ and R.I. Tanner
School of Aerospace, Mechanical and Mechatronic Engineering,
The University of Sydney, NSW 2006, Australia
Submitted to EABE, 13 September 2004; revised 21 December 2004
∗Corresponding author: Telephone +61 2 9351 7151, Fax +61 2 9351 7060, E-mail
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Abstract This paper presents an effective high order boundary integral equation method
(BIEM) for the solution of biharmonic equations. All boundary values including geome-
tries are approximated by high order radial basis function networks (RBFNs) rather than
the conventional low order Lagrange interpolation schemes. For a better quality of ap-
proximation, the networks representing the boundary values and their derivatives are con-
structed by using integration processes. Prior conversions of network weights into nodal
variable values are employed in order to form a square system of equations. Numerical
results show that the proposed BIEM attains a great improvement in solution accuracy,
convergence rate and computational efficiency over the linear- and quadratic-BIEMs.
Keywords: indirect radial basis function networks; biharmonic equations; boundary inte-
gral equation methods.
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1 Introduction
Many engineering problems, for example, viscous fluid flows and thin plate bending prob-
lems, are governed by biharmonic equations. Consequently, considerable effort has been
devoted to the solution of biharmonic equations for many decades. Since exact analyses
of such problems are usually very difficult, a variety of numerical methods have been
developed.
The numerical analysis of biharmonic boundary value problems can be accomplished
by a number of techniques: finite difference methods (FDMs), finite element methods
(FEMs), boundary integral equation methods (BIEMs) and other methods. The BIEM
has certain advantages over the domain type solution methods. For example, the system of
equations obtained by BIEMs is much smaller than the system obtained by FDMs/FEMs.
Furthermore, in solving homogeneous equations, it may require discretization only on the
boundary. Once the boundary solutions are available, the solutions at any interior point
can be obtained by direct evaluation.
Biharmonic boundary value problems have been studied extensively by BIEMs; a great
number of publications are available in the literature (e.g. [1-12]). A boundary integral
formulation for biharmonic boundary value problems consists of two equations in order to
deal with four variables at each point on the boundary (two of them being unknown). A
pair of boundary integral equations (BIEs) can be formed by a) the BIE of a biharmonic
equation (standard BIE) and its normal derivative (hypersingular BIE) [2-4]; b) BIEs of
two harmonic equations [5,6,10,12] or c) BIEs of a biharmonic equation and a harmonic
equation [7,11]. The first formulation involves the complexity and high-order singularity
of the kernel functions, leading to the great difficulty of numerical treatments while the
last two formulations involve only weakly and CPV singular integrals, whose evaluations
can be carried out by using logarithmic Gaussian quadrature/Telles transformation and
constant potential hypothesis, respectively [13-15]. In contrast to the second formulation,
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no volume integrals occur in the third formulation for solving homogeneous biharmonic
equations.
In the context of function approximation/interpolation, the use of high order Lagrange
polynomials does not guarantee to yield a better quality of approximation. High order
schemes can induce some oscillations between nodal function values that may have no
relation at all to the behaviour of the “true” function [16]. In the conventional BIEMs, low
order Lagrange interpolation schemes such as constant, linear or quadratic elements are
usually employed to represent the variations of boundary values. To obtain a high degree
of accuracy, dense meshes are thus required in most cases. For the solution of boundary
integral equations of the first kind, the use of large numbers of boundary elements can
lead to ill-conditioned systems of algebraic equations.
Radial basis function networks (RBFNs) have become one of the main fields of research
in numerical analysis [17]. The construction of an RBFN, in its most basic forms, con-
sists of three layers: the input layer, the hidden layer and the output layer. There is a
large class of RBFs whose interpolation matrices are always solvable provided that the
data points are all different [18]. It has been proved that RBFNs have the property of
universal approximation [19,20]. Another important point is the fact that the dimension
of the hidden space is directly related to the capacity of the network to approximate
a smooth input-output mapping [17]. According to the Cover theorem, the higher the
dimension of the hidden space, the more accurate the approximation will be [21]. Theo-
retically, high order RBFNs can represent any continuous function to a prescribed degree
of accuracy using relatively low numbers of data points. However, in practice, due to
the lack of general mathematical theories for determining the optimal values of network
parameters (e.g. the positions and the widths of RBFs), it is still difficult to achieve such
a universal approximation. A function and its derivatives can be approximated by the
direct RBFN (DRBFN) approach based on a differentiation process [22] or by the indirect
RBFN (IRBFN) approach based on an integration process [23]. From an approximation
4
theoretic point of view, the approximating functions are expected to be much smoother
through the integration process and therefore IRBFNs can have higher approximation
power than DRBFNs. Numerical results have shown that the IRBFN approach performs
better than the DRBFN approach and element-based approximation schemes in terms of
accuracy and convergence rate [23].
For conventional BIE methods, the numerical procedure involves a subdivision of the
boundary into a number of small elements. On each element, the geometry and the vari-
ables are assumed to have certain low order shapes such as constant, linear or quadratic
ones. First order derivatives of the boundary quantities/the slope of approximate repre-
sentations are discontinuous across elements. In the proposed BIE method, the boundary
is subdivided into a number of large segments over each of which the geometry is smooth
and the prescribed boundary conditions are of the same type. IRBFNs are employed to
represent the variations of boundary quantities along these segments. They thus allow
first and higher order derivatives to be continuous on large regions. Each segment is sim-
ply discretized by a set of discrete points rather than a set of intervals/elements. However,
due to the fact that the construction of IRBFNs is based on an integration process, there
are extra network weights (integration constants), leading to a non-square interpolation
matrix.
It is different from previous works dealing with potential and viscous fluid flow problems
that are governed by Poisson and the Navier-Stokes equations [24,25], prior conversions of
the sets of network weights into the sets of nodal variable values are employed here in order
to form a square system of equations that can be solved using Gaussian elimination. The
linear- and quadratic-BIEMs are also considered to provide the basis for the assessment of
the presently proposed IRBFN-BIEM. For the biharmonic Dirichlet problems of the first
kind with non-smooth geometries and boundary conditions of complex shapes, it will be
shown that the iterative decoupled approach has advantages over the usual non-iterative
coupled approach with regard to the capability of preventing large fluctuations in the
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computed boundary solutions. The proposed IRBFN-BIEM is found to be far superior
to the conventional low order BIEMs for all numerical test cases.
The remainder of the paper is organized as follows. A brief review of the integral rep-
resentation due to the coupled biharmonic and harmonic equations is given in section 2.
Section 3 presents the indirect RBFN approach. The proposed IRBFN-BIEM is described
in section 4 and then verified through a number of examples including some thin plate
bending problems in section 5. Section 6 gives some concluding remarks.
2 Boundary integral equations for the biharmonic
boundary value problem
Consider the biharmonic equation defined on a bounded domain Ω in the plane for a
function v(x, y),
∇4v = p(x, y), (1)
where p(x, y) is a known function of position. This equation is coupled with a set of
prescribed boundary conditions to constitutive the biharmonic boundary value problem.
The biharmonic boundary value problems can be formulated in the form of boundary
integral equations. A formulation due to the combined biharmonic and harmonic equa-
tions [7] is adopted here. By introducing a new variable u = ∇2v, the boundary integral
equations of two equations,
∇2u = p(x, y), (2)
∇4v = p(x, y), (3)
6
can be written as
C(P )u(P ) +
∫
Γ
∂GH(P,Q)
∂nu(Q)dΓ =
∫
Γ
GH(P,Q)∂u(Q)
∂ndΓ
−
∫
Ω
GH(P,Q)p(Q)dΩ, (4)
C(P )v(P ) +
∫
Γ
∂GH(P,Q)
∂nv(Q)dΓ =
∫
Γ
GH(P,Q)∂v(Q)
∂ndΓ
−
∫
Γ
(
∂GB(P,Q)
∂nu(Q) − GB(P,Q)
∂u(Q)
∂n
)
dΓ −
∫
Ω
GB(P,Q)p(Q)dΩ, (5)
where P is the source point, Q the field point, Γ the piecewise smooth boundary of Ω,
C(P ) the free term coefficient which is 1 if P is an interior point, 1/2 if P is a point
on the smooth boundary and θ2π
if P is a corner (θ the internal angle of the corner in
radians), n the outward normal to the boundary, GH and GB the harmonic and biharmonic
fundamental solutions, whose forms respectively are
GH =1
2πln
(
1
r
)
, (6)
GB =1
8πr2
[
ln
(
1
r
)
+ 1
]
, (7)
in which r = ‖P − Q‖. For viscous fluid flows, the variables v and u represent the
streamfunction and vorticity, respectively while for thin plate bending problems, they
are the deflection and “bending moment”, respectively. The variable u has no physical
meaning inside the plate domain, however it may be equal to the bending moment on the
boundary.
3 Indirect RBFNs
In the present work, only two dimensional (2D) problems are considered. In view of
the fact that the BIEM allows the reduction of the problem dimension by one, only
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IRBFNs for a function and its derivatives in 1D domain need to be utilized. The ith
order IRBFN, IRBFN-i, is defined as an RBFN approximation scheme in which the ith
order derivative is decomposed into RBFs and then integrated i times. To represent the
k order derivative, the order of IRBFNs should be equal to or greater than k. The effect
of the order of IRBFNs on the accuracy of approximation has been investigated in [26].
Solving the governing integral equations (4)-(5) requires the approximation of functions:
v(s), ∂v(s)/∂n, u(s) and ∂u(s)/∂n, and two first derivatives: ∂x(s)/∂s and ∂y(s)/∂s.
The use of a second order, one order higher than the minimum requirement, seems to be
sufficient and IRBFN-2 will be employed here
d2f(s)
ds2=
m∑
i=1
w(i)g(i)(s), (8)
df(s)
ds=
∫ m∑
i=1
w(i)g(i)(s)ds + C1 =m+1∑
i=1
w(i)H(i)[1] (s), (9)
f(s) =
∫ m+1∑
i=1
w(i)H(i)[1] ds + C2 =
m+2∑
i=1
w(i)H(i)[0] (s), (10)
where superscripts denote the elements of a set of neurons, m is the number of radial
basis functions, g(i)mi=1 the set of RBFs, w(i)m
i=1 the set of network weights to be
found, H(i)[.]
mi=1 new basis functions obtained from integrating the radial basis function
g. For convenience of presentation, the integration constants which are unknowns here
and their associated known basis functions (polynomial) on right hand sides of (9)-(10)
are also denoted by the notations w(i) and H(i)[.] , respectively, but with i > m.
Among RBFs, multiquadrics (MQ) are ranked the best in terms of accuracy [27] and also
possess exponential convergence with respect to the refinement of spatial discretization
[28,29]. In the present work, the MQ is implemented and hence the basis functions g and
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H(i)[.] take the forms as
g(i)(s) =√
(s − c(i))2 + a(i)2, (11)
H(i)[1] (s) =
(s − c(i))
2A +
a(i)2
2B, (12)
H(i)[0] (s) =
(
−a(i)2
3+
(s − c(i))2
6
)
A +a(i)2(s − c(i))
2B, (13)
where c(i)mi=1 is the set of RBF centres, a(i)m
i=1 the set of RBF widths,
A =√
(s − c(i))2 + a(i)2 and B = ln(
(s − c(i)) +√
(s − c(i))2 + a(i)2)
. To make the train-
ing process simple, the centres and widths of RBFs are chosen in advance. For the for-
mer, the set of centres is chosen to be the same as the set of collocation points, i.e.
c(i)mi=1 ≡ s(i)n
i=1 with n = m, while for the latter, the following relation is used
a(i) = βd(i), (14)
where β is a positive scalar and d(i) is the minimum of distances from the ith center to
its neighbours.
In the present study, the set of network weights w(i)m+2i=1 is converted into the set of
function values f(s(i))ni=1 in order to form a square system of equations that can be
solved by Gaussian elimination. The process is as follows. The evaluation of (10) at the
set of collocation points s(i)ni=1 results in
f(s(1))
f(s(2))
...
f(s(n))
=
H(1)[0] (s(1)) · · · H
(m)[0] (s(1)) s(1) 1
H(1)[0] (s(2)) · · · H
(m)[0] (s(2)) s(2) 1
· · ·
H(1)[0] (s(n)) · · · H
(m)[0] (s(n)) s(n) 1
w(1)
w(2)
...
w(m+2)
. (15)
The obtained system (15) for the unknown vector of network weights can be solved using
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general linear least squares
w(1)
w(2)
...
w(m+2)
=
H(1)[0] (s(1)) · · · H
(m)[0] (s(1)) s(1) 1
H(1)[0] (s(2)) · · · H
(m)[0] (s(2)) s(2) 1
· · ·
H(1)[0] (s(n)) · · · H
(m)[0] (s(n)) s(n) 1
−1
f(s(1))
f(s(2))
...
f(s(n))
, (16)
or
w = H−1[0] f , (17)
where H−1[0] is the Moore-Penrose pseudoinverse and the dimensions of w, H−1
[0] and f are
(m + 2) × 1, (m + 2) × n and n × 1, respectively. As can be seen from (17), the network
weights are expressed in terms of the function values f (i)ni=1. By substituting (17) into
(8)-(10), the function f and its derivatives at an arbitrary point s will be computed by
f(s) =
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
f (1)f (2) · · · f (n)
]T
, (18)
df(s)
ds=
[
H(1)[1] (s) · · · H
(m)[1] (s) 1 0
]
H−1[0]
[
f (1)f (2) · · · f (n)
]T
, (19)
d2f(s)
ds2=
[
g(1)(s) · · · g(m)(s) 0 0
]
H−1[0]
[
f (1)f (2) · · · f (n)
]T
. (20)
Since all basis functions in (18)-(20) are available in analytic forms, the present indirect
RBFN formulation is truly meshless.
4 The IRBFN-BIEM algorithm
The boundary integral equations (4)-(5) governing biharmonic boundary value problems
involve weakly and CPV singular integrals. In the present work, the former is evaluated
by Telles quadratic transformation [15] while the latter is treated using the non-CPV-
singular form of BIEs [30]. Equations (4)-(5) without the CPV singular integrals can be
10
written as
∫
Γ
∂GH(P,Q)
∂n(u(Q) − u(P )) dΓ =
∫
Γ
GH(P,Q)∂u(Q)
∂ndΓ,
−
∫
Ω
GH(P,Q)p(Q)dΩ, (21)
∫
Γ
∂GH(P,Q)
∂n(v(Q) − v(P )) dΓ =
∫
Γ
GH(P,Q)∂v(Q)
∂ndΓ
−
∫
Γ
(
∂GB(P,Q)
∂nu(Q) − GB(P,Q)
∂u(Q)
∂n
)
dΓ −
∫
Ω
GB(P,Q)p(Q)dΩ. (22)
The kernels GB(P,Q) and ∂GB(P,Q)∂n
have no singularities; their integrals can be evaluated
by using appropriate Gaussian quadrature formulae.
In the case of non-homogeneous equations, a function p(x, y) generates the volume in-
tegrals. Since the function p is known, these volume integrals do not introduce any
new unknowns. Several techniques have been developed for the evaluation of volume in-
tegrals, including cell integration approach, particular solution technique [31], multiple
reciprocity method (MRM) [32] and dual reciprocity method (DRM) [33]. The present
study is concerned with the enhancement of evaluating boundary integrals by using an
effective interpolation scheme based on IRBFNs. For convenience, all volume integrals in
the work presented here are converted into the boundary integrals with the help of the
MRM,∫
Ω
G[i]f [i]dΩ =∞∑
j=i
∫
Γ
(
∂G
∂n
[j+1]
f [j] − G[j+1] ∂f
∂n
[j])
dΓ, (23)
where the index i indicates a volume integral which occurs in the boundary integral
equation, f [j+1](x, y) = ∇2f [j](x, y), and G[j+1] and ∂G[j+1]/∂n are the (j + 1)th order
11
fundamental solutions to the potential problems, whose forms for 2D problems are
G[j+1] =1
2πr2(j+1)(Aj+1 ln
1
r+ Bj+1), (24)
∂G[j+1]
∂n=
1
2πr2(j+1)−1
[(
2(j + 1) ln1
r− 1
)
Aj+1 + 2(j + 1)Bj+1
]
∂r
∂n, (25)
Aj+1 =Aj
4(j + 1)2, (26)
Bj+1 =1
4(j + 1)2
(
Aj
j + 1+ Bj
)
, (27)
A0 = 1, B0 = 0, (28)
where j = 0, 1, 2, · · · .
• For a volume integral in (21): i = 0, j ≥ 0
j = 0, G[0] = GH , f [0] = p,
j = 1, G[1] = GB, f [1] = ∇2p, · · · ,
• For a volume integral in (22): i = 1, j ≥ 1
j = 1, G[1] = GB, f [1] = p, · · · .
The procedural flow chart is summarized as follows
• Divide the boundary into a number of segments over each of which the boundary is
required to be smooth and to have the same type of prescribed boundary conditions.
Clearly, the size of segments in the proposed method can be chosen to be much
larger than the size of elements in conventional BIEMs. As a result, the IRBFN
approximation scheme allows first and higher order derivatives of the boundary
quantities to be continuous on larger regions than the case of using constant, linear
and quadratic elements. Furthermore, each segment here is simply represented by
12
a number of discrete points, leading to an easy process of numerical modelling.
The boundary integrals in (21)-(22) are replaced by the summations of integrals
corresponding to a series of segments;
• Approximate all boundary values including geometries on each segment by IRBFNs,
e.g. for the variables
v(s) =
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
v(1) v(2) · · · v(n)
]T
,
∂v(s)
∂n=
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
∂v(1)
∂n∂v(2)
∂n· · · ∂v(n)
∂n
]T
,
u(s) =
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
u(1) u(2) · · · u(n)
]T
,
∂u(s)
∂n=
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
∂u(1)
∂n∂u(2)
∂n· · · ∂u(n)
∂n
]T
,
and for the geometry
x(s) =
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
x(1) x(2) · · · x(n)
]T
,
y(s) =
[
H(1)[0] (s) · · · H
(m)[0] (s) s 1
]
H−1[0]
[
y(1) y(2) · · · y(n)
]T
,
∂x(s)
∂s=
[
H(1)[1] (s) · · · H
(m)[1] (s) 1 0
]
H−1[0]
[
x(1) x(2) · · · x(n)
]T
,
∂y(s)
∂s=
[
H(1)[1] (s) · · · H
(m)[1] (s) 1 0
]
H−1[0]
[
y(1) y(2) · · · y(n)
]T
,
dΓ =
√
(
∂x
∂s
)2
+
(
∂y
∂s
)2
ds,
where s is the independent variable, and v(i)ni=1, ∂v(i)/∂nn
i=1, u(i)n
i=1, ∂u(i)/∂nni=1,
x(i)ni=1 and y(i)n
i=1 are the sets of nodal variable values.
• Substitute the IRBFNs representing the boundary values into the boundary integral
equations (21)-(22);
• Discretize the system by collocating BIEs at the boundary nodes and then carry
13
out numerical integrations in the local coordinate system;
• Solve the obtained system of algebraic equations using the coupled or decoupled
approach, depending on the geometry and the type of boundary conditions as will
be discussed in the following section;
• Evaluate the solution at selected interior points;
• Output the results.
5 Numerical examples
A number of examples are presented in this section for the purpose of demonstrating
the effectiveness of the proposed method. The linear- and quadratic-BIEMs are also
considered to provide the basis for the assessment of the IRBFN-BIEM. To describe the
multi-valued normal derivatives ∂v/∂n and ∂u/∂n at a corner, two extreme points on each
segment are shifted into the segment by the amount of a quarter of the length between
two nodes for the IRBFN-BIEM code while discontinuous elements are employed for the
linear- and quadratic-BIEM codes.
In the following test cases, the width of the ith RBF is simply chosen to be the minimum
distance from the ith centre to neighbouring centres (a(i) = βd(i) = d(i)). The accuracy
of numerical solution produced by an approximation scheme is measured via the norm of
relative errors of the solution as follows
Ne =
√
∑nt
i=1 [f0((x, y)(i)) − f((x, y)(i))]2
∑nt
i=1 f0((x, y)(i))2, (29)
where nt is the number of test points, (x, y)(i) is the ith test point, f and f0 are the cal-
culated and exact solutions, respectively. Another important measure is the convergence
14
rate of the solution with respect to the refinement of spatial discretization
Ne(s) ≈ γsα = O(sα) (30)
in which s is the average centre spacing, and α and γ are the exponential model’s param-
eters. Given a set of observations, these parameters can be found by general linear least
squares.
In the present study, of particular interest are the prescribed boundary conditions of the
form: v; ∂2v/∂n2 (the second biharmonic Dirichlet problem) and v; ∂v/∂n (the first
biharmonic Dirichlet problem).
5.1 Boundary conditions given in terms of v and u (BC1)
In this case, each integral equation has its own boundary condition. The integral equa-
tions (21) and (22) are weakly coupled and hence they can be solved separately. An
approximation to the variable ∂u/∂n is first determined by solving (21). The obtained
solution is then utilized to solve (22) for the variable ∂v/∂n.
15
5.1.1 Example 1: a benchmark test problem (BTP), BC1, rectangular do-
main
Consider the homogeneous biharmonic equation ∇4v = 0; the exact solutions are given
by
v =1
2x (sin x cosh y − cos x sinh y) , (31)
∂v
∂x=
1
2(sin x cosh y − cos x sinh y) +
1
2x (cos x cosh y + sin x sinh y) , (32)
∂v
∂y=
1
2x (sin x sinh y − cos x cosh y) , (33)
∂v
∂n=
∂v
∂xnx +
∂v
∂xny, (34)
u = cos x cosh y + sin x sinh y, (35)
∂u
∂x= cos x sinh y − sin x cosh y, (36)
∂u
∂y= sin x cosh y + cos x sinh y, (37)
∂u
∂n=
∂u
∂xnx +
∂u
∂xny, (38)
where nx and ny denote two components of the outward normal unit vector in the x and
y directions, respectively. This is a typical benchmark test problem (BTP), where all
boundary data are nonzero [9].
A unit square domain is considered. The boundary is divided into four segments cor-
responding to four edges of the domain. Ten uniform discretizations of the boundary,
namely 3 × 4, 5 × 4, · · · , 21 × 4 (number of nodes per segment × number of segments),
are employed to study this problem. A set of 26 × 4 test points is used to compute the
error norm Ne of the boundary solutions for all discretizations. The obtained results
are displayed in Figure 1. Results by the linear- and quadratic-BIEMs are also included
for comparison. The proposed method yields the most accurate results, followed by the
quadratic-BIEM and linear-BIEM. At the first four coarse discretizations, the conver-
16
gence rates are O(s1.76), O(s2.70) and O(s3.48) for linear-, quadratic- and IRBFN-BIEMs,
respectively.
5.1.2 Example 2: a simply supported square plate with a concentrated load
at the centre
For this problem, the volume integrals, which are generated by the concentrated force
F = 1 at the centre O, are easily evaluated as follows
∫
Ω
GH(P,Q)p(Q)dΩ =
∫
Ω
GH(P,Q) (F (O)δ(O,Q)) dΩ = F (O)GH(P,O), (39)∫
Ω
GB(P,Q)p(Q)dΩ =
∫
Ω
GB(P,Q) (F (O)δ(O,Q)) dΩ = F (O)GB(P,O). (40)
The analytical solution of this problem can be found in Timoshenko and Woinowsky-
Krieger [34]. Three uniform boundary discretizations, namely 3×4, 5×4 and 7×4, and a
test set of 30× 4 nodes are employed to study mesh convergence. The linear-, quadratic-
and IRBFN-BIEMs yield the convergence rates of O(s1.92), O(s2.05) and O(s4.97), and the
error norms at the finest discretization of 0.012, 0.0052 and 2.42e − 4, respectively.
5.1.3 Example 3: simply supported square plates under a uniform load and
hydrostatic pressure
A simply supported plate of dimension [0, 200] × [0, 200] cm2 with a uniform load q is
considered here. Analytical solutions are available for this problem [34]. The parameters
of the problem are
E = 2.1 × 106kg/cm2, q = 0.5kg/cm2,
h = 10cm, ν = 0.3, D = Eh3/[12(1 − ν2)],
17
where E is the Young’s modulus, ν the Poisson’s ratio, h the plate thickness and D the
flexural rigidity. The source function becomes
p(x, y) = q/D = constant.
For this simple form of the source function, the series in (23) becomes a finite summation.
With the help of the MRM, the volume integrals in (21)-(22) are transformed into the
following boundary integrals
∫
Ω
GHpdΩ =
∫
Ω
G[0]pdΩ =
∫
Γ
(
∂G[1]
∂np − G[1] ∂p
∂n
)
dΓ = p
∫
Γ
∂G[1]
∂ndΓ, (41)
∫
Ω
GBpdΩ =
∫
Ω
G[1]pdΩ =
∫
Γ
(
∂G[2]
∂np − G[2] ∂p
∂n
)
dΓ = p
∫
Γ
∂G[2]
∂ndΓ. (42)
The boundary is divided into 4 segments corresponding to four edges of the domain.
Table 1 summarizes the results on the boundary obtained by the proposed method and
the linear-BIEM [6]. With the same boundary discretization of 9×4, the proposed method
yields more accurate results than the linear-BIEM.
To further demonstrate the effectiveness of the proposed method for the second biharmonic
Dirichlet problems, a simply supported unit square plate under hydrostatic pressure is
considered here. Using only coarse discretizations, accurate results are obtained. For
example, with 5 × 4 boundary nodes, the dimenionless deflections along the horizontal
centreline are u(0.25, 0.5) = 0.00131, u(0.5, 0.5) = 0.00203, u(0.6, 0.5) = 0.00202 and
u(0.75, 0.5) = 0.00162; they are in good agreement with the values reported in [34]:
0.00131, 0.00203, 0.00201 and 0.00162, respectively.
18
5.2 Boundary conditions given in terms of v and ∂v/∂n (BC2)
In this case, all boundary conditions for (22) v and ∂v/∂n are provided (i.e. this equation
is overprescribed on the boundary) while there are no boundary conditions at all for (21)
(i.e. this equation can have many solutions). Equations (21) and (22) form a coupled
pair of BIEs for the variables u and ∂u/∂n and they must be solved simultaneously. A
biharmonic problem with the prescribed boundary conditions v and ∂v/∂n is seen to
provide a good means of testing and validating numerical methods.
It is known that the order of continuity of the dependent variables required by the inverse
statement (BIEM) is less than that required by the weak form (FEM) and the strong
form (FDM). In BIEMs, the dependent variables along the boundary are required to be
square integrable (i.e. they are allowed to be discontinuous at discrete points but finite
throughout the region) [14]. Numerical results will show that any small oscillations in
the computed boundary solution u greatly affect the accuracy of the overall boundary
solutions. For the case of smooth geometries, the non-iterative coupled approach can be
applied directly. However, special care needs to be paid to the case of non-smooth geome-
tries. For some test problems, it is found that the use of iterative decoupled approach to
generate a computational boundary condition for u is necessary in order to prevent large
fluctuations in the computed boundary solutions.
5.2.1 Example 4: BTP, BC2, circular domain (smooth geometry)
The benchmark test problem is considered again, but with a circular domain of radius
R. To avoid the transfinite diameter geometry R = 1 for which the boundary integral
equations (4) and (5) might not have a unique solution [9,35], the radius is chosen to be
R = 2.
The boundary is equally divided into two segments. A number of uniform discretizations
19
(15×2, 20×2, · · · , 40×2) and a test set of 50×2 points are employed along the boundary.
The coupled approach is employed, where two coupled equations (21)-(22) are solved
simultaneously for the whole set of variables u, ∂u/∂n. Again, the IRBFN boundary
results are more accurate than the linear and quadratic ones by several orders of magnitude
as displayed in Figures 2. Figure 3 presents an enlargement of the computed solutions
at some region on the boundary. It should be pointed out that the linear case performs
better than the quadratic case. The reason could be that the boundary solution u obtained
by the quadratic-BIEM is less smooth than that obtained by the linear-BIEM (only C0
continuity provided here for the boundary quantity u between elements as noted earlier),
resulting in larger fluctuations in the boundary solution ∂u/∂n for the former (Figure
3). The boundary solutions converges apparently as O(s1.78), O(s1.15) and O(s2.43) for
linear-, quadratic- and IRBFN-BIEMs, respectively. In this case, increasing the order of
polynomial interpolation leads to a decrease in accuracy. In general, the use of high order
Lagrange schemes needs to be cautious.
Although the boundary solutions obtained by the quadratic-BIEM are less accurate than
those obtained by the linear-BIEM, the former performs better than the latter in the
computation of internal solutions. For example, using the mesh of 40 boundary nodes,
the error norms of the solution u computed at 80 uniformly distributed interior points are
2.0e− 2, 3.3e− 3 and 3.1e− 4 for linear-, quadratic- and IRBFN-BIEMs, respectively. It
appears that spurious oscillations in the computed quadratic-BIEM solutions may cancel
together during the process of evaluating integrals along the boundary.
5.2.2 Example 5: BTP, BC2, rectangular domain (non-smooth geometry)
So far, in dealing with BC2, the proposed method has been verified in a smooth geometry
that gives rise to no new difficulties. The case of a non-smooth geometry involving corners
is further studied here. The benchmark test problem is considered again, with the domain
20
of interest being a square of dimension [−2, 2] × [−2, 2]. The boundary is divided into
four segments. A number of uniform boundary discretizations, 7× 4, 11× 4, 17× 4, 21× 4
and 27 × 4, are employed.
The problem is first solved by the coupled approach. To describe derivative boundary
conditions at a corner, two nodes which meet at the corner are shifted inside the two
segments/elements [36]. The condition numbers of the final systems of equations are from
1.49e4 to 1.10e6 for the above range of discretizations. Figure 4 presents the variations
of the computed solutions u and ∂u/∂n along the boundary including corner regions
(s = 0, s = 4, s = 8, s = 16). Due to the effect of corners, large oscillations appear
especially for linear and quadratic cases.
It is instructive to note that all BIE methods yield accurate internal solutions despite
fluctuations in the boundary solutions. For example, at the discretization of 27 × 27
(i.e. 27 × 4 boundary nodes and 25 × 25 interior nodes), the error norms of the internal
solution v are 0.0054, 3.30e − 4 and 1.11e − 5 for linear-, quadratic- and IRBFN-BIEMs,
respectively. It appears that the evaluation of integrals along the boundary cancels out
spurious oscillations.
From Figure 4, it can be seen that a small amount of noise in the boundary solution
u corresponds to large noise in the boundary solution ∂u/∂n. Any improvement in the
approximation of u is expected to result in a great improvement in the boundary solution
∂u/∂n. Based on this observation, the present work will attempt to employ an alternative
way of obtaining the boundary solution u. This boundary condition is generated through
an iterative process in which two equations are solved separately at each iteration (the
decoupled approach) rather than directly solving the coupled system of equations for the
whole set of variables as previously used (the coupled approach). The process is as follows
1. Initialize the boundary conditions for u and ∂u/∂n;
21
2. Compute the internal solutions ∂v/∂x and ∂v/∂y by carrying out derivatives on the
BIE (5) for interior points;
3. Differentiate the results obtained from the previous step using linear, quadratic or
IRBFN approximation in order to update the boundary condition for u according
to
u =∂2v
∂x2+
∂2v
∂y2;
4. Solve the BIE (21) to update the boundary condition for ∂u/∂n;
5. Check for convergence;
6. If not yet converged, repeat from step 2
7. If converged, stop.
The exact value of the function u at four corners can be found using the prescribed bound-
ary conditions. These values are utilized during the iterative process. For all methods,
the solutions are significantly improved (Figure 5). In the case of quadratic-BIEM, the
boundary solutions converge as O(s0.2473) and O(s0.8664) for the coupled and decoupled
approaches, respectively; at the finest mesh, their error norms are 10.4216 and 0.0464. In
the case of IRBFN-BIEM, the corresponding values are O(s0.4199), O(s1.2387), 0.1662 and
0.0052. Regarding matrix rank, the decoupled approach yields considerably better condi-
tion numbers of the system matrices than the coupled approach; the condition numbers of
the IRBFN-BIEM are from 33.95 to 142.66 for the range of discretizationes used. Results
for the internal solutions are presented in Table 2. The proposed method attains greater
accuracy than the quadratic-BIEM. Fast convergence rates are obtained for both IRBFN-
and quadratic-BIEMs. In this test case, the decoupled approach is far superior to the
coupled approach. In other words, by iteratively generating a computational boundary
condition for u, one can avoid spurious oscillatory behaviour in the computed boundary
solutions.
22
5.2.3 Example 6: a clamped rectangular plate under a central load
The bending of a rectangular plate with clamped edges under a concentrated load F at
the centre is analyzed here. In contrast to the previous example, all boundary data BC2
are zero. The governing integral equations (21)-(22) are thus reduced to
∫
Γ
∂GH(P,Q)
∂n(u(Q) − u(P )) dΓ =
∫
Γ
GH(P,Q)∂u(Q)
∂ndΓ
−
∫
Ω
GH(P,Q)p(Q)dΩ, (43)
∫
Γ
(
∂GB(P,Q)
∂nu(Q) − GB(P,Q)
∂u(Q)
∂n
)
dΓ +
∫
Ω
GB(P,Q)p(Q)dΩ = 0, (44)
which involve the boundary quantities u and ∂u/∂n only. In this simple case (BC2 =
0), numerical results show that no fluctuations are observed in the computed boundary
conditions by using the coupled approach. Both coupled and decoupled approaches yield
smooth boundary solutions.
To compare with the quadratic-BIEM results using 16 quadratic elements obtained by
Karami et al [7], the equivalent boundary discretization of 9 × 4 is employed. The de-
coupled approach is utilized here. Results of the dimensionless deflection at the centre vc
and the maximum dimensionless bending moment at the fixed ends umax are displayed
in Table 3 for various values of b/a in which a and b are the lengths of the sides. Com-
pared with the analytical results [34], good agreement is achieved. For vc, both numerical
methods yield highly accurate results while for umax, the proposed method yields greater
accuracy than the quadratic-BIEM.
5.3 Numerical efficiency
Apart from the above comparison of accuracy and convergence rate, the efficiency of
IRBFN- and linear-BIEMs are examined numerically in this section through the solution
23
of the example in section 5.2.1. All BIEM codes are written in MATLAB language (version
6.5R13 by The MathWorks, Inc.); they are run on a 999MHz Pentium PC. The present
linear-BIEM program is similar to the one produced by Brebbia and Dominguez [36]. The
most time-consuming part is the construction of the system matrix. It is reiterated that
both methods are based on the same flow diagram and their final systems of equations
have the same size. The main distinguishing feature between the two methods is that
the number of elements (segments) in the IRBFN-BIEM is significantly less than that
in the linear-BIEM. Hence, the loop (a FOR statement) to cover the entire boundary is
much shorter with the former. By taking into account the effect of vectorization, the
IRBFN-BIEM code achieves a great improvement in computational efficiency over the
linear-BIEM as shown in Table 4.
6 Conclusion
This paper reports a new high order BIE method for the analysis of biharmonic problems.
Unlike the conventional BIEMs, all boundary values in the governing integral equations
including geometries are represented by indirect RBFNs. Prior conversions of the sets
of network weights into the sets of nodal variable values are employed in order to form
a square system of equations that can be solved by Gaussian elimination. For the first
biharmonic Dirichlet problems with non-smooth geometries and boundary conditions of
complex shapes, the iterative decoupled approach is better than the non-iterative coupled
approach with regard to the capability of preventing large fluctuations in the computed
boundary solutions. Numerical results show that the performance of the proposed method
is far superior to that of the linear and quadratic-BIEMs.
Acknowledgements N. Mai-Duy wishes to thank the University of Sydney for
a Sesqui Postdoctoral Research Fellowship. We would like to thank the referees for their
helpful comments.
24
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28
Table 1: Example 3: the boundary solutions using 9 × 4 boundary nodes.
∂v∂n
× 10−3 (error %)
Method (x = 25, y = 0) (x = 50, y = 0) (x = 75, y = 0) (x = 100, y = 0)Analytical -0.1146 -0.2048 -0.2613 -0.2804IRBFN-BIEM -0.1150(0.35%) -0.2051(0.15%) -0.2616(0.11%) -0.2807(0.11%)Linear-BIEM -0.126 (9.95%) -0.222 (8.40%) -0.276 (5.63%) -0.298 (6.28%)
D∂u∂n
× 102 (error %)
Analytical 0.1959 0.2813 0.3243 0.3376IRBFN-BIEM 0.1961(0.10%) 0.2812(0.04%) 0.3244(0.03%) 0.3376(0.00%)Linear-BIEM 0.2072(5.77%) 0.2828(0.53%) 0.3277(1.05%) 0.3394(0.53%)
29
Table 2: Example 5: the error norm of the internal solutions.
Method 7 × 7 11 × 11 17 × 17 21 × 21 27 × 27Ne(v)
Quadratic 0.0965 0.0358 0.0111 0.0061 0.0029IRBFN 0.0457 0.0075 0.0017 8.28e-4 3.66e-4
Ne(u)
Quadratic 0.0749 0.0272 0.0093 0.0055 0.0029IRBFN 0.0182 0.0038 9.96e-4 5.31e-4 2.53e-4
30
Table 3: Example 6: the dimensionless deflection at the centre vc and the maximumdimensionless bending moment at the fixed ends umax using 16 quadratic elements/9× 4boundary nodes. Note that a and b are the lengths of the sides.
vc umax
b/a Quadratic IRBFN analytical Quadratic IRBFN analytical
1.0 0.0056 0.0056 0.0056 0.1266 0.1257 0.12571.2 0.0065 0.0065 0.0065 0.1502 0.1492 0.14901.6 0.0071 0.0071 0.0071 0.1671 0.1652 0.16512.0 0.0073 0.0072 0.0072 0.1708 0.1675 0.1674
31
Table 4: Example 4: total CPU times used to obtain the boundary solutions at 100 testnodes and the internal solution at 80 uniformly distributed points.
No. of Linear-BIEM IRBFN-BIEM
boundary nodes No. of elements Time (s) No. of elements Time (s)
30 30 5.8 2 0.940 40 8.4 2 1.150 50 11.4 2 1.560 60 14.7 2 1.970 70 18.3 2 2.380 80 22.5 2 2.8
32
10−2
10−1
100
10−5
10−4
10−3
10−2
10−1
100
linearquadraticIRBFN
Boundary node spacing
Ne
Figure 1: Example 1: accuracy and convergence rate.
33
10−1
10−4
10−3
10−2
10−1
100
linearquadraticIRBFN
Boundary node spacing
Ne
Figure 2: Example 4: accuracy and convergence rate.
34
a) Linear
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
exactapproximate
s
u
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−6
−5
−4
−3
−2
−1
0
exactapproximate
s
∂u/∂
nb) Quadratic
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
exactapproximate
s
u
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−6
−5
−4
−3
−2
−1
0
exactapproximate
s
∂u/∂
n
c) IRBFN
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
exactapproximate
s
u
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6−6
−5
−4
−3
−2
−1
0
exactapproximate
s
∂u/∂
n
Figure 3: Example 4, 15×2 boundary nodes: zoom in on the computed boundary solutionsu and ∂u/∂n.
35
a) Linear
0 2 4 6 8 10 12 14 16−50
−40
−30
−20
−10
0
10
20
30
40
50
ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−800
−600
−400
−200
0
200
400
600
800
ExactApproximate
s
∂u/∂
nb) Quadratic
0 2 4 6 8 10 12 14 16−15
−10
−5
0
5
10
15
20ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−250
−200
−150
−100
−50
0
50
100
150
200
250ExactApproximate
s
∂u/∂
n
c) IRBFN
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
∂u/∂
n
Figure 4: Example 5, 11×4 boundary nodes: effect of the corner on the boundary solutionsin the case of using non-iterative coupled approach.
36
a) Linear
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
∂u/∂
nb) Quadratic
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
∂u/∂
n
c) IRBFN
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
u
0 2 4 6 8 10 12 14 16−6
−4
−2
0
2
4
6ExactApproximate
s
∂u/∂
n
Figure 5: Example 5, 11×4 boundary nodes: effect of the corner on the boundary solutionsin the case of using iterative decoupled approach.
37
